Having snow on the mountains isn’t going to give you an ice rink, curling rink, bobsled track and all the other various buildings.
With the exception of the bobsled track, many cities already have the facilities to host the Winter Olympics without having to spend $5-50 Billion. They need to stop picking places where it is warm in the winter. I am sure a lot of the $50 billion Russia spent did not end up in the bobsled track. More likely in Putin’s Swiss bank account.
Denver did not want the 1970 winter Olympics because they did not want to bankrupt the city. They already had the mountains and climate.
Several cold climate cities like Salt Lake and Calgary have been successful in hosting the Olympics. Even small towns like Lake Placid ,St. Moritz and Squaw Valley have done OK.
Structures can be built anywhere. However, you can not replace the Swiss/French/Austria Alps or the US/Canadian Rocky Mountains for cold weather and natural snow in February.